Power — Revision Notes

⚡ 30-Second Revision

Definition: — Rate of doing work or energy transfer.
Average Power: — $P_{avg} = W/t$
Instantaneous Power: — $P_{inst} = dW/dt = \vec{F} \cdot \vec{v}$
SI Unit: — Watt (W). $1,\text{W} = 1,\text{J/s}$
Other Units: — Kilowatt ($1,\text{kW} = 10^3,\text{W}$), Horsepower ($1,\text{hp} \approx 746,\text{W}$)
Energy Unit (related): — Kilowatt-hour ($1,\text{kWh} = 3.6 \times 10^6,\text{J}$)
Efficiency: — $\eta = \frac{P_{out}}{P_{in}} \times 100\%$
Power for lifting: — $P = mgv$ (at constant velocity)
Area under P-t graph: — Represents Work Done ($W = \int P dt$)
Maximum Power: — When $\vec{F}$ and $\vec{v}$ are parallel ($\theta = 0^circ$).
Zero Power: — When $\vec{F}$ and $\vec{v}$ are perpendicular ($\theta = 90^circ$) or $\vec{v}=0$.

2-Minute Revision

Power is the rate at which work is done or energy is transferred. It's a scalar quantity, measured in Watts (W), where $1,\text{W} = 1,\text{J/s}$. There are two main types: average power, calculated as total work divided by total time ($P_{avg} = W/t$), and instantaneous power, which is the power at a specific moment.

Instantaneous power is crucially given by the dot product of force and velocity ($P = vec{F} cdot vec{v}$). This formula highlights that only the component of force parallel to the direction of motion contributes to power.

If force and velocity are perpendicular, power is zero. Efficiency ($eta$) is another key concept, defined as the ratio of useful output power to total input power ($eta = P_{out}/P_{in}$), always less than 100% for real machines due to energy losses.

Remember that kilowatt-hour (kWh) is a unit of energy, not power. For problems involving lifting objects at constant velocity, power is simply $mgv$. The area under a Power-time graph gives the total work done.

5-Minute Revision

Power is a measure of how quickly work is performed or energy is transformed. It's distinct from work or energy, which quantify the total amount. The SI unit is the Watt (W), equivalent to one Joule per second ($1,\text{J/s}$).

We differentiate between average power, $P_{avg} = W/t$, which is the total work over a time interval, and instantaneous power, $P_{inst} = dW/dt$, which is the power at a specific moment. The most versatile formula for instantaneous power is $P = vec{F} cdot vec{v}$, where $vec{F}$ is the force and $vec{v}$ is the velocity.

This implies that power is maximized when force and velocity are in the same direction ($cos 0^circ = 1$) and zero when they are perpendicular ($cos 90^circ = 0$).

Example: A $50,\text{kg}$ person climbs a $3,\text{m}$ high staircase in $5,\text{s}$. What is their average power output? (Take $g = 10,\text{m/s}^2$) Solution: Work done $W = mgh = (50,\text{kg})(10,\text{m/s}^2)(3,\text{m}) = 1500,\text{J}$. Average Power $P_{avg} = W/t = 1500,\text{J} / 5,\text{s} = 300,\text{W}$.

Efficiency ($eta$) is crucial for real-world applications, defined as $eta = (P_{out}/P_{in}) \times 100\%$. It accounts for energy losses, typically as heat. For instance, if a motor has $80$ efficiency and needs to deliver $400,\text{W}$ of output power, the input power required would be $P_{in} = P_{out}/eta = 400, ext{W} / 0.

8 = 500, ext{W}$. Remember that kilowatt-hour (kWh) is a unit of energy, not power, representing$3.6 imes 10^6, ext{J}$. Graphical problems might ask for the area under a Power-time graph, which represents the total work done.

Always pay attention to units and the specific type of power (average or instantaneous) requested.

Prelims Revision Notes

1
Definition of Power: — Power ($P$) is the rate at which work ($W$) is done or energy ($E$) is transferred. $P = dW/dt = dE/dt$.
2
Average Power: — $P_{avg} = \frac{\Delta W}{\Delta t} = \frac{\text{Total Work Done}}{\text{Total Time Taken}}$.
3
Instantaneous Power: — $P_{inst} = \vec{F} \cdot \vec{v} = Fv \cos\theta$, where $\theta$ is the angle between force $\vec{F}$ and velocity $\vec{v}$.

* If $\vec{F} \parallel \vec{v}$ (same direction), $\theta = 0^circ$, $P = Fv$ (maximum power). * If $\vec{F} \perp \vec{v}$ (perpendicular), $\theta = 90^circ$, $P = 0$ (zero power). * If $\vec{F}$ and $\vec{v}$ are opposite, $\theta = 180^circ$, $P = -Fv$ (negative power, energy removed).

1
Units of Power:

* SI Unit: Watt (W). $1,\text{W} = 1,\text{J/s}$. * Common Multiples: Kilowatt ($1,\text{kW} = 10^3,\text{W}$), Megawatt ($1,\text{MW} = 10^6,\text{W}$). * Other Unit: Horsepower ($1,\text{hp} \approx 746,\text{W}$). (For NEET, use $746,\text{W}$ unless specified).

1
Kilowatt-hour (kWh): — This is a unit of energy, not power. $1,\text{kWh} = 1,\text{kW} \times 1,\text{h} = 1000,\text{W} \times 3600,\text{s} = 3.6 \times 10^6,\text{J}$.
2
Efficiency ($eta$): — For any machine, $\eta = \frac{\text{Useful Output Power}}{\text{Total Input Power}} = \frac{P_{out}}{P_{in}}$. Also, $\eta = \frac{\text{Useful Output Energy}}{\text{Total Input Energy}}$. Always $\eta < 1$ (or $100$) for real machines.
3
Power in Lifting Objects: — If an object of mass $m$ is lifted vertically at a constant velocity $v$, the power required is $P = Fv = (mg)v = mgv$.
4
Graphical Interpretation: — The area under a Power-time (P-t) graph gives the total work done or energy transferred: $W = \int P dt$.
5
Variable Force: — If force is variable, work done $W = \int F dx$. Then $P_{avg} = W/t$.
6
Key Distinction: — Power is the *rate* of work/energy, while work/energy is the *total amount*.

Vyyuha Quick Recall

Power Is Fast Velocity. (P = Fv) - Helps remember the instantaneous power formula. Also, Watts Just Seconds (W = J/s) for the unit definition.